Definition of the elastic forces in the finite-element absolute nodal coordinate formulation and the floating frame of reference formulation

The equivalence of the finite-element formulations used in flexible multibody dynamics is the focus of this investigation. This equivalence will be used to address several fundamental issues related to the deformations, flexible body coordinate systems, and the geometric centrifugal stiffening effect. Two conceptually different finite-element formulations that lead to exact modeling of the rigid body dynamics will be used. The first one is the absolute nodal coordinate formulation in which beams and plates can be treated as isoparametric elements. This formulation leads to a constant and symmetric mass matrix and highly nonlinear elastic forces. In this study, it is demonstrated that different element coordinate systems which are used for the convenience of describing the element deformations lead to similar results as the element size is reduced. In particular, two element frames are used; the pinned and the tangent frames. The pinned frame has one of its axes passing through two nodes of the element, while the tangent frame is rigidly attached to one of the ends of the element. Numerical results obtained using these two different frames are found to be in good agreement as the element size decreases. The relationship between the coordinates used in the absolute nodal coordinate formulation and the floating frame of reference formulation is presented. This relationship can be used to obtain the highly nonlinear expression of the strain energy used in the absolute nodal coordinate formulation from the simple energy expression used in the floating frame of reference formulation. It is also shown that the source of the nonlinearity is due to the finite rotation of the element. The result of the analysis presented clearly demonstrates that the instability observed in high-speed rotor analytical models due to the neglect of the geometric centrifugal stiffening is not a problem inherent to a particular finite-element formulation. Such a problem can only be avoided by considering the known linear effect of the geometric centrifugal stiffening or by using a nonlinear elastic model as recently demonstrated. Fourier analysis of the solutions obtained in this investigation also sheds new light on the fundamental problem of the choice of the deformable body coordinate system in the floating frame of reference formulation. Another method for formulating the elastic forces in the absolute nodal coordinate formulation based on a continuum mechanics approach is also presented.